A medium is called anisotropic if the propagation velocity of seismic waves varies with the direction of propagation.
Transverse Isotropy (TI) is a particular type of anisotropy, as described by L. Thomsen in 1986 in the article “Weak elastic anisotropy” published in the magazine Geophysics, 51, 1954-1966, for which there exists an axis of rotational symmetry, while the propagation velocity is invariant for propagation directions in a plane perpendicular to the axis of symmetry.
More particularly, the invention relates to a method for seismic P-wave modelling to generate a seismic image of a subsurface formation that is represented in a Cartesian coordinate frame as an inhomogeneous transversely isotropic acoustic medium with a tilted symmetry axis of variable direction.
Such a method is known from International patent applications WO2009/092025 and WO2010/014379. These patent publications disclose methods for seismic P-wave modelling in Vertical Transversely Isotropic (VTI) media as well as in Tilted TI media. Both patent publications refer to a method disclosed in 2009 by Fletcher, Du and Fowler in an article titled “Reverse time migration in tilted transversely isotropic (TTI) media” published in the magazine Geophysics, 74, WCA179-WCA187, where a non-zero shear velocity is introduced, while retaining a coupled system of two second-order scalar differential equations. In the seismic literature, such an approach was first presented for VTI media at the EAGE Conference (Rome, 2008) and published by Du, Fletcher and Fowler in extended abstract H033, titled “A new pseudo-acoustic wave equation for VTI media”.
The articles published by Du, Fletcher and Fowler deal with a stability problem of different nature than the stability problem solved by the method according to the invention. By introducing a positive shear velocity Vs, Du et al. relieve the physical constraint ε−δ≧0 and can deal with a range of negative ε−δ, depending on the choice of Vs. The resulting discretization schemes generally do not satisfy the principles of self-adjointness accomplished by the method according to the invention. Although the approach proposed by Fletcher, Du and Fowler seems to help achieving numerical stability, synthetic experiments for Tilted TI models show that the method proposed by Fletcher, Du and Fowler can fail to yield stable results.
Another approach of modelling acoustic P-waves in a TTI medium, which is closely related to the present invention, was presented at the SEG Conference (Houston, 2009) by Zhang and Zhang and published in expanded abstract SPMI 2.4 titled “A stable TTI reverse time migration and its implementation”. Zhang and Zhang apply similar stability concepts (based on self-adjointness) as in the present invention, but Zhang and Zhang's wave equations miss out several cross-coupling terms of relevant stress components. Moreover, their discretization scheme differs from that of the present invention, in which discrete first- and second-order difference operators are combined.
It is an object of the present invention to provide an improved and more accurate method for seismic P-wave modelling to generate a seismic image of a subsurface formation that is represented in a Cartesian coordinate frame as an inhomogeneous transversely isotropic (TI) acoustic medium with a tilted symmetry axis of variable direction.